Namespaces
Variants
Actions

Difference between revisions of "Additive theory of ideals"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (\nsubseteq)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
<!--
 +
a0107101.png
 +
$#A+1 = 49 n = 0
 +
$#C+1 = 49 : ~/encyclopedia/old_files/data/A010/A.0100710 Additive theory of ideals
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
A branch of modern algebra. Its principal task is to represent any [[Ideal|ideal]] of a ring (or of another algebraic system) as the intersection of a finite number of ideals of special type (primary, tertiary, primal, uniserial, etc.). The type of the representation is so chosen that: 1) for any ideal there exists a representation, in other words, some  "existence"  theorem holds; 2) the representations chosen must be unique apart from certain limitations or, in other words, some  "uniqueness"  theorem must hold. The fundamental principles of the additive theory of ideals were introduced in the 1920s and the 1930s by E. Noether [[#References|[1]]] and W. Krull [[#References|[2]]].
 
A branch of modern algebra. Its principal task is to represent any [[Ideal|ideal]] of a ring (or of another algebraic system) as the intersection of a finite number of ideals of special type (primary, tertiary, primal, uniserial, etc.). The type of the representation is so chosen that: 1) for any ideal there exists a representation, in other words, some  "existence"  theorem holds; 2) the representations chosen must be unique apart from certain limitations or, in other words, some  "uniqueness"  theorem must hold. The fundamental principles of the additive theory of ideals were introduced in the 1920s and the 1930s by E. Noether [[#References|[1]]] and W. Krull [[#References|[2]]].
  
All special features of the additive theory of ideals are clearly manifested in the case of rings. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a0107101.png" /> be a Noetherian ring, i.e. an associative ring with the maximum condition for ideals. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a0107102.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a0107103.png" />, then there exists a largest ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a0107104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a0107105.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a0107106.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a0107107.png" />. This ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a0107108.png" /> is known as the primary radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a0107109.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071010.png" />) and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071011.png" />. An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071013.png" /> is said to be primary if for any two ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071016.png" />, the condition
+
All special features of the additive theory of ideals are clearly manifested in the case of rings. Let $  R $
 +
be a Noetherian ring, i.e. an associative ring with the maximum condition for ideals. If $  A $
 +
is an ideal of $  R $,  
 +
then there exists a largest ideal $  N $
 +
of $  R $
 +
for which $  N  ^ {k} \subseteq A $
 +
for some integer $  k \geq  1 $.  
 +
This ideal $  N $
 +
is known as the primary radical of $  A $(
 +
in $  R $)  
 +
and is denoted by $  \mathop{\rm pr} (A) $.  
 +
An ideal $  Q $
 +
of $  R $
 +
is said to be primary if for any two ideals $  A $
 +
and $  B $
 +
of $  R $,  
 +
the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071017.png" /></td> </tr></table>
+
$$
 +
AB \subseteq Q , A \nsubseteq Q  \Rightarrow  B \subseteq  \mathop{\rm pr} ( Q )
 +
$$
  
is satisfied. The intersection theorem is valid for primary ideals: The intersection of two primary ideals having the same primary radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071018.png" /> is itself a primary ideal with radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071019.png" />. This theorem is used to prove an existence theorem: If the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071020.png" /> is commutative, then for any ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071021.png" /> there exists a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071022.png" /> as the intersection of a finite number of primary ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071023.png" />:
+
is satisfied. The intersection theorem is valid for primary ideals: The intersection of two primary ideals having the same primary radical $  P $
 +
is itself a primary ideal with radical $  P $.  
 +
This theorem is used to prove an existence theorem: If the ring $  R $
 +
is commutative, then for any ideal $  A \neq R $
 +
there exists a representation of $  A $
 +
as the intersection of a finite number of primary ideals $  A _ {i} $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
= A _ {1} \cap \dots \cap A _ {n} ,
 +
$$
  
such that none of the ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071025.png" /> contains the intersection of the other ones, and such that the primary radicals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071026.png" /> are pairwise different. Such representations are known as non-contractible or primarily reduced [[#References|[1]]], [[#References|[4]]]. The uniqueness theorem holds for such representations: If (1) holds and
+
such that none of the ideals $  A _ {i} $
 +
contains the intersection of the other ones, and such that the primary radicals $  \mathop{\rm pr} ( A _ {i} ) $
 +
are pairwise different. Such representations are known as non-contractible or primarily reduced [[#References|[1]]], [[#References|[4]]]. The uniqueness theorem holds for such representations: If (1) holds and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
= B _ {1} \cap \dots \cap B _ {m}  $$
  
is a second primarily-reduced representation of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071028.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071032.png" />, provided the ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071033.png" /> are suitably renumbered.
+
is a second primarily-reduced representation of the ideal $  A $
 +
of the ring $  R $,  
 +
then $  m = n $
 +
and $  \mathop{\rm pr} ( A _ {i} ) = \mathop{\rm pr} ( B _ {i} ) $
 +
for $  1 \leq  i \leq  n $,  
 +
provided the ideals $  B _ {i} $
 +
are suitably renumbered.
  
 
The additive theory of ideals of Noetherian commutative rings (the classical additive theory of ideals) has found numerous applications in various branches of mathematics.
 
The additive theory of ideals of Noetherian commutative rings (the classical additive theory of ideals) has found numerous applications in various branches of mathematics.
  
If the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071034.png" /> is non-commutative, the above-mentioned existence theorem is no longer valid, but the uniqueness and intersection theorems still hold. This is why, ever since the 1930s, repeated attempts have been made to find a generalization of classical primarity to the non-commutative case such that the existence theorem, too, remains valid. Such a generalization has in fact been found [[#References|[4]]], and is known as tertiarity (cf. [[Tertiary ideal|Tertiary ideal]]). It was subsequently shown that, within certain natural limitations, tertiarity is the only  "good"  generalization of the concept of primarity [[#References|[6]]], [[#References|[7]]], [[#References|[8]]].
+
If the ring $  R $
 +
is non-commutative, the above-mentioned existence theorem is no longer valid, but the uniqueness and intersection theorems still hold. This is why, ever since the 1930s, repeated attempts have been made to find a generalization of classical primarity to the non-commutative case such that the existence theorem, too, remains valid. Such a generalization has in fact been found [[#References|[4]]], and is known as tertiarity (cf. [[Tertiary ideal|Tertiary ideal]]). It was subsequently shown that, within certain natural limitations, tertiarity is the only  "good"  generalization of the concept of primarity [[#References|[6]]], [[#References|[7]]], [[#References|[8]]].
  
 
During the 1960s the additive theory of ideals developed further within the framework of lattice theory, of systems with fractions and of multiplicative systems [[#References|[4]]], [[#References|[5]]], [[#References|[6]]]; this stimulated the development of the additive theory of ideals for non-associative rings, normal divisors of a group and submodules of a module.
 
During the 1960s the additive theory of ideals developed further within the framework of lattice theory, of systems with fractions and of multiplicative systems [[#References|[4]]], [[#References|[5]]], [[#References|[6]]]; this stimulated the development of the additive theory of ideals for non-associative rings, normal divisors of a group and submodules of a module.
Line 23: Line 70:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Noether,  "Idealtheorie in Ringbereichen"  ''Math. Ann.'' , '''83'''  (1921)  pp. 24–66</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Krull,  "Idealtheorie in Ringen ohne Endlichkeitsbedingung"  ''Math. Ann.'' , '''101'''  (1929)  pp. 729–744</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Lesieur,  R. Croisot,  "Algèbre noethérienne noncommutative" , Gauthier-Villars  (1963)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K. Murata,  "Additive ideal theory in multiplicative systems"  ''J. Inst. Polytechn. Osaka City Univ. (A)'' , '''10''' :  2  (1959)  pp. 91–115</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.A. Andrunakievich,  Yu.M. Ryabukhin,  "The additive theory of ideals in systems with residuals"  ''Math. USSR-Izv.'' , '''1''' :  5  (1967)  pp. 1011–1040  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31'''  (1967)  pp. 1057–1090</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.A. Riley,  "Axiomatic primary and tertiary decomposition theory"  ''Trans. Amer. Math. Soc.'' , '''105'''  (1962)  pp. 177–201</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  I.M. Goyan,  Yu.M. Ryabukhin,  "On the axiomatic additive theory of Riley ideals"  ''Mat. Issl.'' , '''2''' :  1  (1967)  pp. 14–25  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  L. Fuchs,  "On primal ideals"  ''Proc. Amer. Math. Soc.'' , '''1'''  (1950)  pp. 1–6</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  ''Itogi Nauk. Algebra Topol. Geom. 1965''  (1967)  pp. 133–180</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Noether,  "Idealtheorie in Ringbereichen"  ''Math. Ann.'' , '''83'''  (1921)  pp. 24–66</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Krull,  "Idealtheorie in Ringen ohne Endlichkeitsbedingung"  ''Math. Ann.'' , '''101'''  (1929)  pp. 729–744</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Lesieur,  R. Croisot,  "Algèbre noethérienne noncommutative" , Gauthier-Villars  (1963)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K. Murata,  "Additive ideal theory in multiplicative systems"  ''J. Inst. Polytechn. Osaka City Univ. (A)'' , '''10''' :  2  (1959)  pp. 91–115</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.A. Andrunakievich,  Yu.M. Ryabukhin,  "The additive theory of ideals in systems with residuals"  ''Math. USSR-Izv.'' , '''1''' :  5  (1967)  pp. 1011–1040  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31'''  (1967)  pp. 1057–1090</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.A. Riley,  "Axiomatic primary and tertiary decomposition theory"  ''Trans. Amer. Math. Soc.'' , '''105'''  (1962)  pp. 177–201</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  I.M. Goyan,  Yu.M. Ryabukhin,  "On the axiomatic additive theory of Riley ideals"  ''Mat. Issl.'' , '''2''' :  1  (1967)  pp. 14–25  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  L. Fuchs,  "On primal ideals"  ''Proc. Amer. Math. Soc.'' , '''1'''  (1950)  pp. 1–6</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  ''Itogi Nauk. Algebra Topol. Geom. 1965''  (1967)  pp. 133–180</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A primary representation is also called a primary decomposition. More generally one has primary decompositions for submodules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071035.png" /> of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071036.png" /> over a Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071037.png" />. This is a representation
+
A primary representation is also called a [[primary decomposition]]. More generally one has primary decompositions for submodules $  N $
 +
of modules $  M $
 +
over a Noetherian ring $  R $.  
 +
This is a representation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071038.png" /></td> </tr></table>
+
$$
 +
= \cap _ { i } Q _ {i}  $$
  
where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071039.png" /> consists of a single prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071040.png" />. (By definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071041.png" /> for a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071042.png" />, the set of prime ideals associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071043.png" />, is the collection of all prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071044.png" /> for which there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071046.png" />). There is also a corresponding uniqueness theorem, stating that there is a reduced decomposition, which of course means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071047.png" /> holds for no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071048.png" /> and that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010710/a01071049.png" /> are all different.
+
where each $  \mathop{\rm Ass} ( M / Q _ {i} ) $
 +
consists of a single prime ideal $  \mathfrak p _ {i} $.  
 +
(By definition $  \mathop{\rm Ass} ( M ) $
 +
for a module $  M $,  
 +
the set of prime ideals associated to $  M $,  
 +
is the collection of all prime ideals $  \mathfrak p $
 +
for which there exists an $  x \in M $
 +
such that $  \mathfrak p = \{ {r \in R } : {r x = 0 } \} $).  
 +
There is also a corresponding uniqueness theorem, stating that there is a reduced decomposition, which of course means that $  \cap _ {j \neq i }  Q _ {j} \subset  Q _ {i} $
 +
holds for no $  i \in I $
 +
and that the $  \mathop{\rm Ass} ( M / Q _ {i} ) = \mathfrak p _ {i} $
 +
are all different.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Algèbre commutative" , Hermann  (1961)  pp. Chapt. 3; 4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Algèbre commutative" , Hermann  (1961)  pp. Chapt. 3; 4</TD></TR></table>

Latest revision as of 20:23, 4 April 2020


A branch of modern algebra. Its principal task is to represent any ideal of a ring (or of another algebraic system) as the intersection of a finite number of ideals of special type (primary, tertiary, primal, uniserial, etc.). The type of the representation is so chosen that: 1) for any ideal there exists a representation, in other words, some "existence" theorem holds; 2) the representations chosen must be unique apart from certain limitations or, in other words, some "uniqueness" theorem must hold. The fundamental principles of the additive theory of ideals were introduced in the 1920s and the 1930s by E. Noether [1] and W. Krull [2].

All special features of the additive theory of ideals are clearly manifested in the case of rings. Let $ R $ be a Noetherian ring, i.e. an associative ring with the maximum condition for ideals. If $ A $ is an ideal of $ R $, then there exists a largest ideal $ N $ of $ R $ for which $ N ^ {k} \subseteq A $ for some integer $ k \geq 1 $. This ideal $ N $ is known as the primary radical of $ A $( in $ R $) and is denoted by $ \mathop{\rm pr} (A) $. An ideal $ Q $ of $ R $ is said to be primary if for any two ideals $ A $ and $ B $ of $ R $, the condition

$$ AB \subseteq Q , A \nsubseteq Q \Rightarrow B \subseteq \mathop{\rm pr} ( Q ) $$

is satisfied. The intersection theorem is valid for primary ideals: The intersection of two primary ideals having the same primary radical $ P $ is itself a primary ideal with radical $ P $. This theorem is used to prove an existence theorem: If the ring $ R $ is commutative, then for any ideal $ A \neq R $ there exists a representation of $ A $ as the intersection of a finite number of primary ideals $ A _ {i} $:

$$ \tag{1 } A = A _ {1} \cap \dots \cap A _ {n} , $$

such that none of the ideals $ A _ {i} $ contains the intersection of the other ones, and such that the primary radicals $ \mathop{\rm pr} ( A _ {i} ) $ are pairwise different. Such representations are known as non-contractible or primarily reduced [1], [4]. The uniqueness theorem holds for such representations: If (1) holds and

$$ \tag{2 } A = B _ {1} \cap \dots \cap B _ {m} $$

is a second primarily-reduced representation of the ideal $ A $ of the ring $ R $, then $ m = n $ and $ \mathop{\rm pr} ( A _ {i} ) = \mathop{\rm pr} ( B _ {i} ) $ for $ 1 \leq i \leq n $, provided the ideals $ B _ {i} $ are suitably renumbered.

The additive theory of ideals of Noetherian commutative rings (the classical additive theory of ideals) has found numerous applications in various branches of mathematics.

If the ring $ R $ is non-commutative, the above-mentioned existence theorem is no longer valid, but the uniqueness and intersection theorems still hold. This is why, ever since the 1930s, repeated attempts have been made to find a generalization of classical primarity to the non-commutative case such that the existence theorem, too, remains valid. Such a generalization has in fact been found [4], and is known as tertiarity (cf. Tertiary ideal). It was subsequently shown that, within certain natural limitations, tertiarity is the only "good" generalization of the concept of primarity [6], [7], [8].

During the 1960s the additive theory of ideals developed further within the framework of lattice theory, of systems with fractions and of multiplicative systems [4], [5], [6]; this stimulated the development of the additive theory of ideals for non-associative rings, normal divisors of a group and submodules of a module.

References

[1] E. Noether, "Idealtheorie in Ringbereichen" Math. Ann. , 83 (1921) pp. 24–66
[2] W. Krull, "Idealtheorie in Ringen ohne Endlichkeitsbedingung" Math. Ann. , 101 (1929) pp. 729–744
[3] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[4] L. Lesieur, R. Croisot, "Algèbre noethérienne noncommutative" , Gauthier-Villars (1963)
[5] K. Murata, "Additive ideal theory in multiplicative systems" J. Inst. Polytechn. Osaka City Univ. (A) , 10 : 2 (1959) pp. 91–115
[6] V.A. Andrunakievich, Yu.M. Ryabukhin, "The additive theory of ideals in systems with residuals" Math. USSR-Izv. , 1 : 5 (1967) pp. 1011–1040 Izv. Akad. Nauk SSSR Ser. Mat. , 31 (1967) pp. 1057–1090
[7] J.A. Riley, "Axiomatic primary and tertiary decomposition theory" Trans. Amer. Math. Soc. , 105 (1962) pp. 177–201
[8] I.M. Goyan, Yu.M. Ryabukhin, "On the axiomatic additive theory of Riley ideals" Mat. Issl. , 2 : 1 (1967) pp. 14–25 (In Russian)
[9] L. Fuchs, "On primal ideals" Proc. Amer. Math. Soc. , 1 (1950) pp. 1–6
[10] Itogi Nauk. Algebra Topol. Geom. 1965 (1967) pp. 133–180

Comments

A primary representation is also called a primary decomposition. More generally one has primary decompositions for submodules $ N $ of modules $ M $ over a Noetherian ring $ R $. This is a representation

$$ N = \cap _ { i } Q _ {i} $$

where each $ \mathop{\rm Ass} ( M / Q _ {i} ) $ consists of a single prime ideal $ \mathfrak p _ {i} $. (By definition $ \mathop{\rm Ass} ( M ) $ for a module $ M $, the set of prime ideals associated to $ M $, is the collection of all prime ideals $ \mathfrak p $ for which there exists an $ x \in M $ such that $ \mathfrak p = \{ {r \in R } : {r x = 0 } \} $). There is also a corresponding uniqueness theorem, stating that there is a reduced decomposition, which of course means that $ \cap _ {j \neq i } Q _ {j} \subset Q _ {i} $ holds for no $ i \in I $ and that the $ \mathop{\rm Ass} ( M / Q _ {i} ) = \mathfrak p _ {i} $ are all different.

References

[a1] N. Bourbaki, "Algèbre commutative" , Hermann (1961) pp. Chapt. 3; 4
How to Cite This Entry:
Additive theory of ideals. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_theory_of_ideals&oldid=18106
This article was adapted from an original article by V.A. Andrunakievich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article